Towards p-adic Quantum Information Theory

Towards p-adic Quantum Information Theory

Seminar author:Vincenzo Parisi

Event date and time:05/18/2023 04:00:pm

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Within the framework of quantum mechanics over a quadratic extension of the non-Archimedean field of p-adic numbers, we provide a general definition of a quantum state relying on a general algebraic approach and on a p-adic model of probability theory. As in the standard complex case, a distinguished set of physical states are related to a notion of trace for a certain class of bounded operators and, in fact, we show that one can define a suitable space of trace class operators in the non-Archimedean setting, as well.  The statistical interpretation of the (new) theory is completed by suitably defining the observables in p-adic quantum mechanics. We show that the self-adjoint-operator-valued measures (SOVMs) – a suitable p-adic counterpart of the POVMs associated with a complex Hilbert space – provide a convenient mathematical tool for this scope. The analogies and the differences with respect to the case of standard quantum mechanics in a complex Hilbert space are analyzed.